Properties of convex polytopes

Convex Polytopes play a central role in Linear Programming and are of interest in topology. This dissertation presents a few combinatorial results about the boundary complexes of Convex Polytopes. The fundamental Balinski's Theorem gives a necessary condition for a graph to be isomorphic to the 1-skeleton of a convex polytope. We prove a generalization of Balinski's Theorem and partially resolve another conjectured generalization. Simple Polytopes form a large and practically important class of polytopes. The properties of simple polytopes preclude the polytopes from having certain vertex-cardinalities. We prove tight bounds on the number of vertices that a simple polytope can have. We also present a few results concerning a graphical characterization of the 1-skeleta of simple polytopes, conjectured by Micha Perles. A related result is the proof of nonexistence of a certain topological triangulation of the 3-ball and solid torus. Finally we investigate how affine subspaces of various dimensions intersect with the boundary complexes of polytopes. Our results yield sharp bounds for the dimension of the subspace that can intersect the relative interiors of all the faces of a fixed dimension. We also construct a counterexample to disprove that for every pair of vertices of a polytope there exists a hyperplane passing through the vertices and containing at least two facets of the polytope in one closed halfspace. The relevant results from the literature used in the arguments (in the dissertation) are summarized at the beginning.